Question : What is the value of $\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}$?
Option 1: 6
Option 2: 9
Option 3: 7
Option 4: 8
Correct Answer: 6
Solution :
Given: The trigonometric expression is $\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}$.
We know that $\sin(90^{\circ}–\theta)=\cos \theta$
⇒ $\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}=\frac{3 \sin (90^{\circ}–32^{\circ})}{\cos 32^{\circ}}+\frac{3 \sin (90^{\circ}–48^{\circ})}{\cos 48^{\circ}}$
$=\frac{3 \cos 32^{\circ}}{\cos 32^{\circ}}+\frac{3 \cos 48^{\circ}}{\cos 48^{\circ}}=3+3=6$
Hence, the correct answer is 6.
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